Thermoelastic device comprising an expansive element formed from a preselected material

ABSTRACT

A thermoelastic device comprising an expansive element is disclosed. The expansive element is formed from a material, which is preselected by calculating a dimensionless constant for the material. The dimensionless constant is indicative of the potential effectiveness of the material in a particular application, such as a micro-electromechanical system.

This is a Continuation of 10/636,259 filed on Aug. 8, 2003

Co-Pending Applications

Various methods, systems and apparatus relating to the present inventionare disclosed in the following co-pending applications filed by theapplicant or assignee of the present invention on May 23, 2000:09/575,197, 09/575,195, 09/575,159, 09/575,132, 09/575,123, 09/575,148,09/575,130, 09/575,165, 09/575,153, 09/575,118, 09/575,131, 09/575,116,09/575,144, 09/575,139, 09/575,186, 09/575,185, 09/575,191, 09/575,145,09/575,192, 09/575,181, 09/575,193, 09/575,156, 09/575,183, 09/575,160,09/575,150, 09/575,169, 09/575,184, 09/575,128, 09/575,180, 09/575,149,09/575,179, 09/575,187, 09/575,155 09/575,133, 09/575,143, 09/575,196,09/575,198 09/575,178, 09/575,164, 09/575,146, 09/575,174, 09/575,163,09/575,168, 09/575,154, 09/575,129 09/575,124, 09/575,188, 09/575,189,09/575,162, 09/575,172, 09/575,170, 09/575,171, 09/575,161, 09/575,141,09/575,125, 09/575,142, 09/575,140, 09/575,190, 09/575,138, 09/575,126,09/575,127, 09/575,158, 09/575,117, 09/575,147, 09/575,152, 09/575,176,09/575,151 09/575,177, 09/575,175 09/575,115, 09/575,114, 09/575,113,09/575,112, 09/575,111, 09/575,108, 09/575,109, 09/575,110, 09/575,182,09/575,173, 09/575,194, 09/575,136, 09/575,119, 09/575,135, 09/575,157,09/575,166, 09/575,134, 09/575,121, 09/575,137, 09/575,167, 09/575,120,09/575,122

The disclosures of these co-pending applications are incorporated hereinby cross-reference.

FIELD OF THE INVENTION

The present invention relates to materials potentially suitable for useas the expansive element in thermoelastic design and to methods forranking the potential relative suitabilities of those materials.

The invention as developed originally as a means of identifying andranking a range of materials that potentially may exhibit superiorproperties for use in the manufacture of microscopic thermal bendactuators for use in micro-electro mechanical systems (MEMS), and willbe described hereinafter with reference to this field. However, it willbe appreciated that the invention is not limited to this particular useand is equally applicable to macroscopic design even though the overalldesign considerations are vastly different and certainly less complex.

BACKGROUND OF THE INVENTION

It is important to clarify that thermoelastic actuation is characterizedusing force, deflection and temperature as opposed to switching, whichis characterized using deflection and temperature rise alone.Macroscopic thermoelastic actuators are typically used as switches thatactivate other more energy efficient actuation systems, however,microscopic thermoelastic actuators are an attractive actuationmechanism for a number of reasons. This includes the down scaling ofcertain physical phenomena. For example, it is possible to fabricatevery thin films that decrease the thermal mass and minimize efficiencylosses. Opposing gravitational and inertial forces become negligible onthe microscopic scale. Other advantages include ease of fabrication(although more complex than simple electrostatic actuators) and thepossibility of low voltage operation. Disadvantages include a lowoperational bandwidth determined by the thermal conductivities ofsubstrate materials—this is more of an advantage for the currentapplication allowing for rapid firing.

A relatively diverse range of output force and deflection values can beobtained by altering actuator geometry. However, the fundamentaloperation of actuation is directly related to the mechanical and thermalproperties of the component materials. Correct material selection inassociation with effective design can result in either a smaller or amore efficient actuator. Such an actuator increases wafer yield and isthus more commercially viable. A more efficient actuator may be batterypowered increasing operation simplicity and negating the requirement forexpensive voltage transformers. An increase in thermal efficiencyimproves the operational firing frequency, and decreases the possibilityof thermal crosstalk. This is especially relevant for arrays of thermalactuators in a micro-cilia device.

However, material selection for MEMS application is not straightforward.Firstly, published thin film properties can vary greatly due todifferent fabrication methods and difficulties associated withexperimentally quantifying material properties on the microscopic scale.Secondly, certain thin films can only be fabricated with certain layerthicknesses because inherent stress can shatter or curl the substratewafer. Thirdly, only certain materials can be used in the fabricationprocess at most fabs as the introduction of a new material cancontaminate machinery.

Progress to Date

Until recently, the only materials regularly used or considered for usein such applications were polysilicon, single crystal silicon. However,the applicant just previously made the surprising discovery thattitanium nitride and titanium boride/diboride exhibited excellentproperties relevant to this application.

Realizing the breakthrough this surprising discovery signified, theapplicant sought to try and identify possible alternatives in order toprovide designers of thermoelastic systems with more choice andflexibility. However, given the lack of available data on their filmproperties for various materials and the fact that empirical testingwith MEMS would be prohibitively expensive, there was clearly a need, orit was at least highly desirable to be able to determine a method ofevaluating materials for this use based solely on the commonly availablemacro material properties.

SUMMARY OF THE INVENTION

It is therefore an ultimate object of one aspect of this invention toidentify a range of alternative materials that will potentially exhibitsuperior properties for use in thermoelastic design and of anotheraspect to provide a means of ranking the potential suitability of agiven range of materials for this same use.

According to a first aspect of the invention there is provided a methodof selecting a material for use as the expansive element in athermoelastic design by deriving an indicator of the material'spotential effectiveness for that use, said method including the step ofcalculating a dimensionless constant εγ for that material in accordancewith the formula: ${ɛ\gamma} = \frac{E\quad\gamma^{2}T}{\rho\quad C}$wherein E is the Young's modulus of the material; γ is the coefficientof thermal expansion; T is the maximum operating temperature, ρ is thedensity and C is the specific heat capacity.

In accordance with a second aspect the invention, in another broad form,also provides a method of manufacturing a thermoelastic element thatincludes at least one expansive element, the method including:

selecting a material for use as the expansive element in thethermoelastic design by deriving an indicator of the material'spotential effectiveness for that use, said method including the step ofcalculating a dimensionless constant εγ for that material in accordancewith the formula: ${ɛ\gamma} = \frac{E\quad\gamma^{2}T}{\rho\quad C}$wherein E is the Young's modulus of the material; γ is the coefficientof thermal expansion; T is the maximum operating temperature, ρ is thedensity and C is the specific heat capacity and selecting the materialon the basis of ε, andmanufacturing the thermoelastic element with the at least one expansiveelement formed of the selected material.

Preferably, the method of selection includes the step of normalizing thedimensionless constant relative to that of silicon to a value ε which isachieved by deriving the value εγ for the material of interest at therelevant temperature value and dividing this by the value of ε obtainedfor silicon at that same temperature.

The relevant maximum operating temperature will depend upon thesurrounding materials and their function but is most commonly theoxidizing temperature or the melting point temperature.

Desirably, the selection method includes the step of eliminating certainmaterials by requiring a pre-determined resistivity range. In onepreferred form this resistivity range is between 0.1 μΩm and 10.0 μΩm.

In accordance with a third aspect of the invention there is provided anexpansive element in a thermoelastic design that is made from anyfunctionally suitable material or combinations of materials selectedfrom a group including:

silicides and carbides of titanium.

In accordance with a fourth aspect of the invention there is provided anexpansive element in a thermoelastic design that is made from anyfunctionally suitable material or combinations of materials selectedfrom a group including:

borides, silicides, carbides and nitrides of tantalum, molybdenum,niobium, chromium, tungsten, vanadium, and zirconium.

In accordance with a fifth aspect of the invention there is provided anexpansive element in a thermoelastic design that is made from anyfunctionally suitable alloy material or combinations of alloy materialsselected from the group including:

borides, silicides, carbides and nitrides of titanium, tantalum,molybdenum, niobium, chromium, tungsten, vanadium, and zirconium.

Preferably the expansive element in a thermoelastic design in accordancewith the third, fourth or fifth aspect of the invention also includesone or more of the following properties:

(a) a resistivity between 0.1 μΩm and 10.0 μΩm;

(b) chemically inert in air;

(c) chemically inert in the chosen ink; and

(d) depositable by CVD, sputtering or other thin film depositiontechnique.

BRIEF DESCRIPTION OF THE DRAWINGS

Derivation of the dimensionless constant ε of the first aspect of theinvention, together with sample applications and examples of derivedvalues of this constant and other properties for a range of materials,will now be described in detail with reference to the accompanyingdrawings in which:

FIG. 1 shows a schematic representation of a thermoelastic actuator;

FIG. 2 shows a plot of longitudinal work versus heat energy for singlematerial clamped/free titanium beam (length 20 μm, thickness 1 μm, width5 μm);

FIG. 3 shows a plot derived from FIG. 2 of expansion efficiency versustemperature efficiency for a clamped/free titanium beam; and

FIG. 4 shows a comparison of mechanical work versus the heat energy ofthermoelastic actuator fabricated from Titanium and Silicon.

DETAILED DESCRIPTION

A non-dimensionalized material actuation efficiency is presented thatassesses the potential application of a material to thermoelasticdesign. The method is based on the material thermal and mechanicalproperties and assists in a structured approach of material selectionfor effective design.

The Material Actuation Efficiency

Actuators are characterized by a combination of deflection, force andoperation temperature in contrast to switches that are characterized byoperation temperature and deflection alone. Fundamental thermoelasticdesign is characterized by the differential longitudinal expansion oftwo bonded layers. Thus, the expansion of isolated unbonded layersdirectly relates to global behavior. A single material beam is used hereto illustrate the material actuation efficiency. The approach isstraightforward and relates to general thermoelastic design. Thederivation assumes that material properties are constant across thethermal range.

Equations 1 to 3 are fundamental thermomechanical equations describingthe behavior of simple single material beam subjected to a quantity ofheat, Q as illustrated in FIG. 1. Equation 1 describes the extension,δL, of a free/free beam and equation 2 describes the reaction force, F,of a clamped/clamped beam.δL=γL₀T   (EQ 1)Where: δL=extension of beam, L₀=original length of beam, T=operationtemperature (temperature rise), and γ=coefficient of thermal expansionof beam.F=AEγT   (EQ2)F=force exerted by beam expansion, A=cross sectional area of beam,E=Young's Modulus.

Q=VρCt   (EQ3)

Where: Q=heat energy input, V=volume of beam, ρ=density, and C=specificheat capacity of beam.

Potential mechanical work is given by equation 4 and is defined as theproduct of the clamped beam force, F, and free beam deflection, δL. Thequadratic relationship between the heat input and output mechanical workfor the simple monolithic beam is shown in FIG. 18.W=FδL   (EQ4)Where: W=mechanical work

Equation 5 describes the non-dimensional thermoelastic actuationefficiency and is formulated as the quotient of the mechanical work andheat energy as described by equations 3 and 4. The efficiency isindependent of geometry and is a primary indication of a material'spotential application to thermoelastic design. The linear relationshipbetween the actuation efficiency and material temperature for the simplebeam is shown in FIG. 3. The graph indicates that high temperatureoperation is desirable for maximum efficiency. The plot is limited bythe applicable operation temperature and therefore, different materialplots are of different lengths. The assumption used in this text is thatthe operation temperature is the material melting point because it isindicative of the operable thermal range. Thus, the material actuationefficiency, ε, is defined as the actuation efficiency at the maximumoperable temperature, T, of that material. The slope of the efficiencycurve is a constant, m_(ε) and is defined in equation 6. The combinationof ε and m_(ε) fully characterize a materials actuation characteristicsnon graphically. $\begin{matrix}{ɛ = {\frac{{Output}\quad{Mechanical}\quad{Work}}{{Heat}\quad{Energy}\quad{Input}} = {\frac{E\quad\gamma^{2}T}{\rho\quad C}\left\lbrack \frac{\left( {N/m^{2}} \right)\left( {{1/{^\circ}}\quad C^{2}} \right)\left( {{^\circ}\quad C} \right)}{\left( {{kg}/m^{3}} \right)\left( {{{Nm}/{{kg}{^\circ}}}\quad C} \right)} \right\rbrack}}} & \left( {{EQ}\quad 5} \right) \\{m_{ɛ} = {\frac{\mathbb{d}ɛ}{\mathbb{d}T} = {\frac{{E\quad\gamma^{2}}\quad}{\rho\quad C}\left\lbrack \frac{{N/m^{2}}\quad{1/{^\circ}}\quad C^{2}}{{{kg}/m^{3}}{{Nm}/{{kg}{^\circ}}}\quad C} \right\rbrack}}} & \left( {{EQ}\quad 6} \right)\end{matrix}$Material Selection

Different thin film materials including materials with extremeproperties (PTFE—high g, Diamond—high E) and compounds from all themajor CVD groups including borides, silicides, nitrides and carbides isshown in Table 2. The efficiency values are scaled according to siliconefficiency values because the inclusion of scaled values greatlysimplifies design equations described in the following text. The scalingor comparison of a material with respect to a reference material is anintegral step in the material selection process. In addition, scalingalso results in a more readable index as illustrated by the followingcomparisons. Silicon is chosen as the reference material because of itspredominance in lithographic fabrication.

Preliminary candidates for thermoelastic actuation can be selectedaccording to efficiencies and slopes, however, it is important to notethat two materials that have identical ε but differing m₆₈ will outputdifferent amounts of work for any constant geometry (see Comparison 1below, different amounts of heat energy are also required). Threeimportant design parameters are defined here as heat input, work outputand volume. A design matrix can be constructed by varying each parameterand can then be used to select suitable materials. The followingcomparisons are used to assemble the design matrix. TABLE 2 MaterialProperties. g E r C m_(e)/m_(r,e) O.T M.P. MN MN KXX R Material 10⁻⁶/°C. GPa kg/m³ J/kg° C. ° C.⁻¹ ° C. ° C. O.T. M.P. W/m · K mWm Aluminum23.1 68.9 2700 897 17.12 657 7.98 231 0.027 Boron Carbide 4.5 454 2520955 4.31 2450 7.49 35 5e4 Chromium diBoride 11.1 540 5600 690 19.42 10002150 13.78 29.62 32 0.18 Chromium diSilicide 5.9 5600 1150 1560 0.8Chromium Carbide 9.9 385 6680 530 12.02 1100 1895 9.38 16.16 19 0.75Chromium Oxide 9.0 102 5210 730 2.45 1000 2603 1.74 4.52 30 13 Copper16.5 110 8940 386 9.79 1085 7.53 398 0.017 Gold 14.2 80 19300 129 7.311064 5.52 315 0.023 Hafnium Carbide 6.3 410 12670 190 7.63 600 3930 3.2421.25 13 0.4-0.6 Hafnium diBoride 7.6 11200 300 1500 3250 51 0.1 HafniumdiSilicide 8030 1100 1700 Hafnium Monocarbide 6.5 424 11940 3890 8 0.5Hafnium Nitride 6.5 13,940 500 3300 17 32 Molybdenum 4.8 343 10200 2513.48 2623 6.48 138 Molybdenum Boride 5 685 7480 530 4.87 1000 2140 3.467.40 27 0.18 Molybdenum Carbide 6.7 530 9120 315 9.34 500 2500 3.3116.56 22 0.57 Molybdenum diSilicide 8.4 450 6240 550 10.44 1700 205012.58 15.17 49 0.7 Nickel 13.4 200 8900 444 10.25 1455 10.58 90.7Niobium diBoride 8.6 650 7210 420 17.91 850 3000 10.80 38.10 0.12 17Niobium diSilicide 8.5 5690 900 2050 0.5 Niobium Carbide 7.4 450 7820290 12.26 650 3500 5.65 30.42 14 0.19 PTFE 220 1.3 2130 1024 32.54 2004.62 140 10e22 Silicon 3.0 162 2330 705 1.00 1410 1410 1 1 149 2300Silicon Carbide 4.7 304 3440 669 3.29 2700 6.30 90 0.5 Tantalum Carbide6.7 510 14500 190 9.37 650 3900 4.32 25.93 23 0.35 Tantalum diBoride 8.5250 12600 250 6.47 850 3090 3.90 14.17 16 0.14 Tantalum diSilicide 9.59080 360 800 2670 0.46 Titanium Carbide 7.4 462 4920 480 12.08 700 31606.00 27.08 17.2 1.55 Titanium diBoride 8.2 575 4450 632 15.51 1400 325315.40 35.78 26.4 0.13 Titanium diSilicide 10.7 270 4100 480 17.72 13001540 16.34 19.35 46 0.145 Titanium Nitride 9.4 600 5450 636 17.25 5002950 6.12 36.10 30 1.35 Tungsten Boride 5.0 790 13100 460 3.70 1000 23652.62 6.20 52 0.19 Tungsten Carbide 5.2 690 15800 200 6.66 500 2780 2.3613.13 29 0.2 Tungsten diSilicide 7.0 300 9750 330 5.15 1200 2165 4.397.91 48 33e10 Vanadium diBoride 7.6 260 5100 670 4.96 600 2430 2.11 8.5442 0.13 Vanadium Carbide 6.7 420 5480 530 7.32 600 2730 3.12 14.18 100.59 Vanadium diSilicide 11.2 5100 1000 1700 25 0.66 Vanadium Nitride8.1 460 6080 630 8.89 450 2170 2.84 13.68 5.2 0.85 Zirconium Carbide 6.3410 6560 250 11.19 600 3440 4.76 27.31 22 0.42 Zirconium diBoride 5.9340 6170 1300 3245 58 0.15 Zirconium diSilicide 8.7 270 4900 1150 160015 0.76 Zirconium Nitride 5.9 500 7350 400 6.68 500 2950 2.37 13.97 100.2-0.3Where:γ = Coefficient of thermal expansion.E = Young's Modulus,ρ = density,C = specific heat capacity,O.T. = Oxidizing temperature,M.T. = Melting Temperature,m_(ε) = Efficiency Slope (normalized to Silicon m_(ε) value, normalizedSilicon value m_((r,ε)) = 0.8865e−06),ε_(c) = Material Index (normalized to Silicon ε value, normalizedSilicon ε_(r) = 1.25e−03),KXX = thermal conductivity, andR = resistivity.Comparison 1

The mechanical work and heat input between a material and silicon for aconstant beam volume is compared. Thus, Comparison 1 calculates themaximum possible relative work and associated relative heat inputrequired due to a direct material substitution. Details of thecomparison for different materials are included in Table 3 which showsthat CVD ceramics are far superior actuator materials than silicon(Table 3 is formulated using melting point and Table 4 is formulatedusing oxidation temperature). Titanium nitride can output 159.3 timesmore the amount of mechanical work than silicon with only 4.41 times theamount of heat input. The factor in equation 8 and the scaled materialefficiency ratio (as included in Table 2) repeatedly occur in thefollowing comparisons illustrating the versatility of the method.$\begin{matrix}{\frac{W_{c}}{W_{r}} = {\frac{ɛ_{c}Q_{c}}{ɛ_{r}Q_{r}} = {\frac{ɛ_{c}}{ɛ_{r}}\left( \frac{\rho_{c}C_{c}T_{c}}{\rho_{r}C_{r}T_{r}} \right)}}} & \left( {{EQ}\quad 7} \right)\end{matrix}$The r subscript denotes the reference material which is silicon in thiscase. The c subscript demotes the compared material. $\begin{matrix}{\frac{Q_{c}}{Q_{r}} = \left( \frac{\rho_{c}C_{c}T_{c}}{\rho_{r}C_{r}T_{r}} \right)} & \left( {{EQ}\quad 8} \right)\end{matrix}$Comparison 2

Different materials increase in temperature by different amounts whensubjected to the same quantity of heat energy for a constant volume. Thematerial volume is scaled relative to the silicon volume according tothe constraints that the same amount of silicon heat energy is input toboth actuators and the compared material attains its operationaltemperature. Thus, the actuation efficiency value remains unchangedbecause it is not a function of volume and the operable temperature isreached (as equation 5 shows). Comparison 2 represents the design casewhere heat and volume are critical factors.

The scaled volume and output mechanical work are calculated usingequations 9 and 10. The volume change is typically implemented bymodifying one geometric dimension, i.e. length, width or thickness.Titanium nitride is capable of 36.1 times the amount of work thatsilicon is capable with the same heat energy input but with only 0.23times the volume. Equation 9 is the inverse of equation 8 and equation10 is simply the scaled efficiency number as included in Table 2.$\begin{matrix}{Q_{r} = {{V_{r}\rho_{r}C_{r}T_{r}} = {Q_{c} = {\left. {V_{c}\rho_{c}C_{c}T_{c}}\Rightarrow\frac{V_{({c,{Qr}})}}{V_{r}} \right. = \frac{\rho_{r}C_{r}T_{r}}{\rho_{c}C_{c}T_{c}}}}}} & \left( {{EQ}\quad 9} \right)\end{matrix}$

The first entry of the bracketed subscript in these equations refers tothe material that the beam is constructed from. The second entry refersto the constraining variable for the described parameter. Forexample−W_((c,Vc))=Mechanical work output from beam constructed ofcompared material with a volume of V_(c). $\begin{matrix}{\frac{W_{({c,{Vc}})}}{W_{({r,{Vr}})}} = {\frac{ɛ_{c}Q_{r}}{ɛ_{r}Q_{r}} = \frac{ɛ_{c}}{ɛ_{r}}}} & \left( {{EQ}\quad 10} \right)\end{matrix}$Comparison 3

The output mechanical work resulting from silicon heat energy forconstant volume beams is compared. The operation temperature andefficiency value for the compared material changes. However, the newefficiency is easily calculated using a multiplicative ratio of the newand old operation temperatures because of the linear relationshipbetween temperature and efficiency (as shown in FIG. 3). The newoperation temperature and work are given by equations 11 and 12. Thiscomparison represents the design case where heat is a criticalparameter.

PTFE will melt when subjected to the input silicon heat value. Titaniumdisilicide outperforms titanium nitride mainly because of the highercomputed operating temperature (Table 3). $\begin{matrix}{Q_{r} = {{V_{r}\rho_{r}C_{r}T_{r}} = {Q_{c} = \left. {V_{c}\rho_{c}C_{c}T_{({c,{Qr}})}}\Rightarrow{T_{r}\left( \frac{\rho_{r}C_{r}}{\rho_{c}C_{c}} \right)} \right.}}} & \left( {{EQ}\quad 11} \right)\end{matrix}$Comparison 4 $\begin{matrix}{\frac{W_{({c,{Qr}})}}{W_{({r,{Qr}})}} = {\frac{ɛ_{({c,{Qr}})}Q_{r}}{ɛ_{r}Q_{r}} = {\frac{T_{({c,{Qr}})}ɛ_{2}}{T_{c}ɛ_{r}} = {\left( \frac{\rho_{r}C_{r}T_{r}}{\rho_{c}C_{c}T_{c}} \right)\frac{ɛ_{c}}{ɛ_{r}}}}}} & \left( {{EQ}\quad 12} \right)\end{matrix}$The material volume is scaled with respect to the silicon volumeaccording to the constraint that the compared material operationtemperature and silicon work are maintained. Thus, if the silicon workvalue is less then the original work then the volume is scaled down.Otherwise the volume is increased as is the case for PTFE or amorphousSilicon Dioxide. The material actuation efficiency reoccurs in thecalculations as an inverse as shown in equation 14

Titanium nitride can output the same amount of work as silicon but witha volume that is less than two orders of magnitude smaller with an inputheat energy that is less than an order smaller. $\begin{matrix}{W_{r} = {{V_{r}E_{c}\gamma^{2}T_{r}^{2}} = {W_{c} = {\left. {V_{c}E_{c}\gamma_{c}^{2}T_{c}^{2}}\Rightarrow\frac{V_{({c,{Wr}})}}{V_{r}} \right. = \frac{E_{r}\gamma_{r}^{2}T_{r}^{2}}{E_{c}\gamma_{c}^{2}T_{c}^{2}}}}}} & \left( {{EQ}\quad 13} \right) \\{\frac{Q_{({c,{Vc}})}}{Q_{({r,{Vr}})}} = {\frac{ɛ_{r}W_{r}}{ɛ_{c}W_{r}} = \frac{ɛ_{r}}{ɛ_{c}}}} & \left( {{EQ}\quad 14} \right)\end{matrix}$Comparison 5

The input heat energy required to output silicon mechanical work forconstant volume beams is compared. The operation temperature and thusefficiency value for the compared material changes. The new efficiencycan be calculated in an identical fashion to that described incomparison 3. The operational temperature and heat input values arecalculated using equations 15 and 16.

The table shows that titanium diSilicide slightly outperforms titaniumnitride whereas both PTFE and silicon dioxide will melt. The CVDceramics are again shown to have the best performance. $\begin{matrix}{W_{r} = {{V_{r}E_{r}\gamma_{r}^{2}T_{r}^{2}} = {W_{c} = {\left. {V_{c}E_{c}\gamma_{c}^{2}T_{c}^{2}}\Rightarrow T_{({c,{Wr}})} \right. = {\left( \frac{\gamma_{r}}{\gamma_{c}} \right)\sqrt{\frac{E_{r}}{E_{c}}}}}}}} & \left( {{EQ}\quad 15} \right) \\\begin{matrix}{\frac{Q_{({c,{Wr}})}}{Q_{({r,{Wr}})}} = \frac{ɛ_{r}W_{r}}{ɛ_{({c,{Qr}})}W_{r}}} \\{= \frac{ɛ_{r}T_{c}}{ɛ_{c}T_{({c,{Qr}})}}} \\{= {\frac{ɛ_{r}T_{c}\gamma_{c}}{ɛ_{c}T_{r}\gamma_{r}}\sqrt{\frac{E_{c}}{E_{r}}}}}\end{matrix} & \left( {{EQ}\quad 16} \right)\end{matrix}$ TABLE 3 Design comparisons for materials included in Table2. Comparisons are done using melting point temperature Comparison 1Comparison 2 Comparison 3 Comparison 4 Comparison 5 Constant Q V, Q W V,W V V_((c,Qr))/ W_((c,Vc))/ W_((c,Qr))/ V_((c,Wr))/ Q_((c,Vc))/Q_((c,Wr))/ Q_(c)/Q_(r) W_(c)/W_(r) V_((r,Qr)) W_((r,Vr)) T_((c,Qr))W_((r,Qr)) V_((r,Vr)) Q_((r,Vr)) T_((c,Wr)) Q_((r,Wr)) Aluminum 0.695.48 1.46 7.98 >Tmelt 0.183 0.125 280.79 0.29 Boron Carbide 2.55 19.060.39 7.49 962.41 2.94 0.053 0.133 561.51 0.58 Chromium diBoride 3.59106.23 0.28 29.62 599.41 8.26 0.009 0.0330 208.73 0.35 Chromium Carbide2.90 46.80 0.35 16.16 654.20 5.58 0.021 0.062 277.16 0.42 Chromium Oxide4.27 19.34 0.23 4.52 608.98 1.06 0.052 0.221 592.32 0.97 Copper 1.6212.18 0.62 7.53 671.18 4.66 0.082 0.132 311.11 0.46 Gold 1.14 6.31 0.875.52 930.29 4.82 0.159 0.181 423.90 0.46 Hafnium Carbide 4.08 86.81 0.2421.25 962.13 5.20 0.012 0.047 422.05 0.44 Molybdenum 2.90 18.78 0.346.48 904.67 2.23 0.053 0.154 605.63 0.67 Molybdenum Boride 3.66 27.090.27 7.40 584.23 2.02 0.037 0.135 411.42 0.70 Molybdenum Carbide 3.1051.36 0.32 16.56 806.23 5.34 0.019 0.061 349.05 0.43 MolybdenumdiSilicide 3.04 46.09 0.33 15.17 674.86 4.99 0.022 0.066 302.14 0.45Nickel 2.48 26.26 0.40 10.58 586.13 4.26 0.038 0.095 284.10 0.48 NiobiumdiBoride 3.92 149.44 0.25 38.10 764.86 9.71 0.007 0.026 245.55 0.32Niobium Carbide 3.43 104.26 0.29 30.42 1021.31 8.88 0.010 0.032 342.970.34 PTFE 0.19 0.87 5.31 4.62 >Tmelt 1.152 0.216 >Tmelt Silicon 1.001.00 1.00 1 1410.00 1.00 1.000 1 1410.00 1.00 Silicon Carbide 2.68 16.910.37 6.30 1006.42 2.35 0.059 0.158 657.00 0.65 Tantalum Carbide 4.64120.27 0.22 25.93 840.70 5.59 0.008 0.038 355.83 0.42 Tantalum diBoride4.20 59.57 0.24 14.17 735.28 3.37 0.017 0.071 400.60 0.54 Titanium 1.707.27 0.59 4.28 984.12 2.52 0.138 0.234 619.87 0.63 Titanium diBoride3.95 141.32 0.25 35.78 823.54 9.06 0.007 0.028 273.81 0.33 TitaniumdiSilicide 1.31 25..32 0.76 19.35 1176.90 14.79 0.040 0.0517 306.22 0.26Titanium Nitride 4.41 159.36 0.23 36.10 668.21 8.18 0.006 0.0277 233.830.35 Tungsten Boride 6.15 38.16 0.16 6.20 384.36 1.01 0.026 0.161 383.101.00 Tungsten Carbide 3.79 49.80 0.26 13.13 732.95 3.46 0.020 0.076394.10 0.54 Tungsten diSilicide 3.01 23.80 0.33 7.91 719.86 2.63 0.0420.126 444.06 0.62 Vanadium diBoride 3.58 30.63 0.28 8.54 677.83 2.380.033 0.117 439.34 0.65 Vanadium Carbide 3.42 48.53 0.29 14.18 797.464.14 0.021 0.071 392.10 0.49 Vanadium Nitride 3.59 49.09 0.28 13.68604.67 3.81 0.020 0.0731 309.91 0.51 Zirconium Carbide 2.44 66.51 0.4127.31 1412.28 11.21 0.015 0.0366 422.05 0.30 Zirconium Nitride 3.7452.32 0.27 13.97 787.80 3.73 0.019 0.0716 408.09 0.52

TABLE 4 Design comparisons for material included in Table 2. Comparisonsare done using oxidation temperature Comparison 1 Comparison 2Comparison 3 Comparison 4 Comparison 5 Constant Q V, Q W V, W VV_((c,Qr))/ W_((c,Vc))/ W_((c,Qr))/ V_((c,Wr))/ Q_((c,Vc))/ Q_((c,Wr))/Q_(c)/Q_(r) W_(c)/W_(r) V_((r,Qr)) W_((r,Vr)) T_((c,Qr)) W_((r,Qr))V_((r,Vr)) Q_((r,Vr)) T_((c,Wr)) Q_((r,Wr)) Vanadium diBoride 0.8851.864 1.13 2.10 >T oxid. 0.326 0.475 439.337 0.648 Vanadium Carbide0.752 2.341 1.33 3.11 >T oxid. 0.26 0.32 392.1 0.49 Vanadium Nitride0.74 2.1 1.34 2.83 >T oxid. 0.289 0.353 309.9 0.513 Zirconium Carbide0.425 2.02 2.35 4.75 >T oxid. 0.301 0.21 422.05 0.299 Zirconium Nitride0.64 1.5 1.57 2.36 >T oxid. 0.405 0.423 408.1 0.518A Thermoelastic Actuator

A hot arm/cold arm actuator is presented in FIG. 1 to illustrate theresults contained in Table 3. Only the steady state solution for aquantity of heat input to the heater is analyzed. The device comprisestwo identical material layers separated by air and connected to eachother at the ends by a thermally non-conductive block. Theforce/deflection characteristics of the output mechanical power can betuned by altering the separation between the two layers. A greaterseparation increases the transverse force but decreases deflection.

Two actuators constructed from titanium and silicon are compared usinggraphed energy results in FIG. 4. Five design comparisons for Titaniumare plotted according to the results contained in Table 3. Therelationship between volumes, mechanical work and heat energy areidentical to those included in Table 3. Titanium volumes are scaledusing length for Comparisons 2 and 4.

Discussion

The combination of five separate material properties is important inassessing a material's potential for thermoelastic design and materialswith one predominant property have been shown to not necessarily be thebest candidate. This is evident in both Table 3 for PTFE (high g) anddiamond (high E). Both gold and copper have high g values but arehindered as good candidates by low E and high r values. Silicon is veryinefficient compared to certain other materials, however, amorphoussilicon dioxide is possibly the most inefficient material of all.

Output mechanical work, input heat energy and actuator volume are threeessential characterizing parameters for thermoelastic design. The designmethod described incorporates these parameters using only materialproperties and provides a structured approach for material selection.The method is versatile because the approach assesses the potential of amaterial using easily calculated comparison ratios. It is important tonote that the approach is a measure of a materials potential and must beused as a tool in conjunction with other appropriate design criteria.For example, criteria such as force/deflection characteristics of theoutput work, material resistivity, environmental ruggedness and materialavailability may be important. The operable temperature range is assumedto be from 0 degrees to the melting point on the Centigrade scalebecause it is indicative of the material thermal range. However, themaximum operable temperature could be different due to oxidation of thematerial or other thermal design constraints. Titanium nitride has closeto the highest actuation efficiency value when melting point is used asa criteria. However, Titanium diSilicide is potentially a bettercandidate for use when oxidation temperature is used. Titanium nitrideis a practical candidate because it is well established as a CMOSbarrier material. The oxidation temperature of TiN can be raised from500° C. to 900° C. by alloying with aluminum. The alloyed material has asymbol (Ti,Al)N.

The actuation efficiency of a simple thermoelastic titanium beam is lowcompared to other actuation mechanisms (less than 1 percent). It istheoretically possible to get a thermoelastic actuation efficiency ofabout 4.5 percent for a simple titanium nitride beam, however, thisvalue typically decreases when the material is implemented in a MEMSdevice due to associated operational losses (for example—thermalconduction into the substrate).

The invention has been described herein by way of example only. Skilledworkers in this field will readily recognize many variations andmodifications which do depart from the spirit and scope of the broadinventive concept.

1. A thermoelastic device comprising an expansive element formed from apreselected material, said material being selected on the basis of ε,wherein εγ is a dimensionless constant for that material in accordancewith the formula:${ɛ\quad\gamma} = \frac{E\quad\gamma^{2}T}{\rho\quad C}$ wherein E isthe Young's modulus of the material; γ is the coefficient of thermalexpansion; T is the maximum operating temperature, ρ is the density andC is the specific heat capacity.
 2. The thermoelastic device of claim 1,wherein the dimensionless constant is normalized relative to that ofsilicon to a value ε, said normalized value being calculated by derivingthe value εγ for the preselected material at the relevant temperaturevalue and dividing this by the value of ε obtained for silicon at thatsame temperature.
 3. The thermoelastic device of claim 1, wherein thepreselected material is selected on the basis of both m_(ε) and ε, where$\begin{matrix}{m_{ɛ} = {\frac{\mathbb{d}ɛ}{\mathbb{d}T} = {\frac{E\quad\gamma^{2}}{\rho\quad C}\quad\left\lbrack \frac{N\text{/}m^{2}1\text{/}{{^\circ}C}^{2}}{{kg}\text{/}m^{3}{Nm}\text{/}{kg}\quad{{^\circ}C}} \right\rbrack}}} & \left( {{EQ}\quad 6} \right)\end{matrix}$
 4. The thermoelastic device of claim 1, wherein thepreselected material is further selected on the basis that it has apre-determined resistivity range.
 5. The thermoelastic device of claim4, wherein the resistivity range is between 0.1 μΩm and 10.0 μΩm.
 6. Thethermoelastic material of claim 1, wherein the preselected material isfurther selected on the basis that it has at least one of the followingproperties: a resistivity between 0.1 μΩm and 10.0 μΩm; chemically inertin air; chemically inert in the chosen ink; or depositable by CVD,sputtering or other thin film deposition technique.
 7. The thermoelasticdevice of claim 1, wherein the preselected material is selected from thegroup including: silicides and carbides of titanium; borides, silicides,carbides and nitrides of tantalum, molybdenum, niobium, chromium,tungsten, vanadium, and zirconium.
 8. The thermoelastic device of claim1, which is a thermal bend actuator.
 9. A micro-electomechanical systemcomprising one or more thermoelastic devices according to claim 8.